$(30$ points$)$ On a cold winter day, a heating unit operates using R$-134$a$.$ The unit is designed to maintain an indoor occupant space at $25C.$ The unit operates using the following processes:

$12$ : adiabatic compression from a saturated vapor at $200$ kPa to $)=(0\mathrm{.}95$

$23$ : constant pressure condensation to a saturated liquid $34$ : constant enthalpy expansion in a throttling valve to the initial pressure

$41$ : constant pressure evaporation to the initial state

$($a$)(20$ points$)$ Determine the quality, temperature, pressure, enthalpy, and entropy at each state in the cycle.

$($b$)(5$ points$)$ Determine the refrigerator COP.

$($c$)(5$ points$)$ If the outdoor tericperature is $-5C,$ determine the Carnot COP of this cycle.

HINT: Carefully consider boundary temperatures which heat transfers through in each process.